We derive a macroscopic model for biofilm formation in a porous medium reactor to investigate the role of suspended bacteria on reactor performance. The starting point is the mesoscopic one-dimensional Wanner–Gujer biofilm model. The following processes are included: hydrodynamics and transport of substrate in the reactor, biofilm and suspended bacteria growth in the pore space, attachment of suspended cells to the biofilm, and detachment of biofilm cells. The mesoscopic equations are up-scaled from the biofilm scale to the reactor scale, yielding a stiff system of balance laws, which we study numerically. We find that suspended bacteria and attachment can have a significant effect on biofilm reactor performance.

We propose and analyse finite volume Godunov type methods based on discontinuous flux for a 2×2 system of non-linear partial differential equations proposed by Hadeler and Kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table. The problem considered here is the so-called partially open table problem where sand is blocked by a wall (of infinite height) on some part of the boundary of the table. The novelty here is the corresponding modification of boundary conditions for the standing and the rolling layers and generalization of the techniques of the well-balancedness proposed in [1]. Presence of walls may lead to unbounded or discontinuous surface flow density at equilibrium resulting in solutions with singularities propagating from the extreme points of the walls. A scheme has been proposed to approximate efficiently the Hamiltonians with the coefficients which can be unbounded and discontinuous. Numerical experiments are presented to illustrate that the proposed schemes detect these singularities in the equilibrium solutions efficiently and comparisons are made with the previously studied finite difference and Semi-Lagrangian approaches by Finzi Vita et al.